Biomedical Engineering Reference
In-Depth Information
Table 2 Input data
Quantity name
Symbol
Value
Unit
Substrate elasticity
E
s
5
kPa
Cell elasticity
E
c
0.5
kPa
Cell radius
R
4
lm
Cell traction force
F
1
lN
Cell death probability
p
0.001
-
Cell division probability
q
0.005
-
Probability of velocity perturbation
p
mp
0
-
Cell mobility coefficient
b
i
0
:
167
10
3
s
1
s
1
Initial relaxation parameter
j
1000
Friction coefficient
l
0.2
-
where M
i
and M
ij
respectively denote the total strain energy density and the
contribution to the strain energy density from the elastic interaction between
neighboring cells. This quantity should be regarded as some energy relative to a
certain energy level or as a potential in order to allow it to have negative values.
For more details, we refer to [
1
]. The data that we use here can be found in
Table
2
.
We show some snapshots of a cell-colony simulation for 'wound healing' in
Fig.
5
. The red dots denote the cells that are moving towards each other as a result
of mechanical pulling and their mechanical sensing. The snapshots at consecutive
times show how the 'wound' closes. Further, we show the 'wound area' versus
time in Fig.
6
. It can be seen that first the 'wound' expands a bit and subsequently
the 'wound' contracts. The curve shows a bit of noise that originates mainly from
the randomness in cell division and cell death. The overall curve looks like a
sigmoid relation. This is confirmed by in-vitro experiments on cell colonies. If one
likes to model angiogenesis and its relation to wound healing, one could use a
cellular automata model for instance and combine this model with the presently
described model.
2.3 The Tissue Scale
In order to be able to perform simulations over larger volumes and areas of tissues,
individual cells are no longer tracked. Instead, cell densities are considered. In
other words, the number of cells per unit volume or area is considered. This
approach gives a system of partial differential equations (PDEs) where densities of
several cell types are considered. In [
29
], among many other studies, a PDE-model
for cutaneous wound healing is considered in terms of tracking the densities of
fibroblasts, endothelial cells (to model angiogenesis), and keratinocytes are con-
sidered. The right-hand side in the above PDE contains a logistic growth term to
account for an increase of cell density towards an equilibrium (i.e. the undamaged
state). Furthermore, the levels of oxygen, VEGF, and extra-cellular matrix are
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